The present invention is generally directed to a method of analyzing signals. More particularly, the invention is directed to a method for extracting parameter information from a time-varying signal.
It is well known that sounds and other mechanical vibration signals typically have actual frequency spectra, especially if the sound persists for even a few cycles of the frequencies involved. A vibrating object will normally produce a fundamental frequency and a sequence of overtones that relate in some manner to the fundamental, often as integer or near-integer multiples. Almost any mechanical system can be described as a collection of such vibrating objects, whether it is a violin string, a human vocal cord, or even a bridge or tall building. If the characteristics of a mechanical system are known well enough, the exact frequency spectrum that it produces can be determined through mechanical modeling. A simple system is shown in FIG. 1.
As shown in FIG. 1, the fundamental frequency is 10.21 Hz and it produces a series of overtones at frequency multiples of 2, 3, 4, 5, and 6. When this mechanical system operates it produces a vibrational signalxe2x80x94in effect a weighted sum of these frequencies and overtonesxe2x80x94which is conventionally analyzed using conventional spectrum analysis. FIGS. 2 and 3 illustrate two of the many possibilities for the vibrational signal as a function of time that might be expected from this example system. Both versions are equally valid, differing only in the phase relationships between the different overtone components.
Conventional spectrum analysis as applied to a sound signal or a recording of a sound signal typically makes use of the Fourier Transform, typically some form of a Discrete Fourier Transform (DFT), often the Fast Fourier Transform (FFT), which is a specific manner of implementing a DFT. The DFT requires that a signal be digitized, that is, that a series of samples of the amplitude of the signal be taken at successive increments of time over a short length of time called the sample time or the sample interval. This series of amplitudes over the sample interval is transformed into a corresponding representation which is called the frequency spectrum.
Typical spectrum analysis applies some form or modification of the DFT to a digitized form of the signal sampled over a time interval. The modification of the basic DFT almost invariably includes some form of windowing. Windowing is a process of pre-multiplying the digitized sample by a set of weights in order to improve the amplitude and frequency resolution of the DFT and has been a central part of spectrum analysis during nearly the entire history of using the method on digital computers.
The lowest frequency represented in the spectrum is (1/sample interval), e.g., if the sample interval is 1 second the lowest frequency is 1 Hz. For a DFT, if the increments of time are evenly spaced, then the frequencies in the spectrum that results from the DFT will also be evenly spaced. In the above example 1 Hz is the first frequency represented, so the frequencies will be 1 Hz apart and the frequency spectrum will have entries for 1 Hz, 2 Hz, 3 Hz, 4 Hz, etc. The number of frequencies thus represented will be one-half the number of points in the sample interval. The DFT frequency spectrum resulting from this operation is shown in FIG. 4.
FIG. 4 is computed directly from the real system spectrum shown in FIGS. 1-3 and is an accurate representation of the spectrum for an evenly spaced DFT. The example chosen is far from being a worst case. There is only a superficial resemblance of the so-called DFT frequency spectrum to the actual frequency spectrum of the simple mechanical system (FIG. 1), with considerable xe2x80x9cleakagexe2x80x9d of the pure frequencies into adjacent frequencies. The visually apparent amplitudes differ substantially and seemingly randomly from the true spectrum. This problem is obvious and serious and was recognized by the early practitioners of digital spectrum analysis.
A single frequency vibrational tone typically causes amplitudes to be detected in several nearby frequencies of the DFT spectrum rather than in just the one or two nearest, due to a phenomenon called sideband leakage. The detected amplitude can vary considerably from that of a known test signal and it is customaryxe2x80x94almost universalxe2x80x94to use windowing to minimize this and force the frequency spectrum peak to be sharper and to more nearly approximate the true amplitude. Windowing usually entails multiplying the sampled values by a sequence of numbers chosen so that the result tapers to zero at either end of the sample interval and is the unaltered signal near the middle of the sample interval.
Windowing does offer considerable improvement in comparison to the above graph, and produces a graph intermediate in appearance between the real spectrum and the DFT spectrum, but leakage and consequent seemingly random variations of amplitude remain. While amplitude information can be partially regained by windowing, phase information is essentially completely lost in conventional spectrum analysis.
To obtain higher precision in the frequency of components, conventional spectrum analysis requires a longer sample intervalxe2x80x94the signal must be sampled for a longer period of time. A longer sample interval does not necessarily improve the estimates of amplitude, however, and phase information is still lost in the analysis. The requirement for longer sample times in order to obtain greater accuracy of frequency determination is a fact of life in the current state of the art. To get approximately 1 Hz accuracy requires a one second sample time, with xc2xd second giving approximate 2 Hz accuracy and 0.1 second sample time giving 10 Hz accuracy. Alternative methods of frequency determination such as the various forms of autocorrelation also require comparatively long sample times and tend to work only in situations where the signal is fairly simple, with a single frequency component or with components spaced well apart in frequency.
Time domain analysis has been used to estimate the frequency, amplitude, and phase of some frequency components of real signals. Here a conventional spectrum analysis is customarily used to provide an initial guess at the frequencies of the various components and the sine/cosine pairs for each frequency are fit to the actual time series data using an iterative scheme to converge to better estimates of the individual components. This approach requires a long sample time for accuracy. Time domain analysis falls apart at short sample times or if there are closely spaced frequencies or if there are frequencies which are not included in the analysis for other reasons.
Several techniques can be made to work when signals are stable enough so that long sample times are not a problem. For example, radio and television transmission signals are designed to be stable and can be very accurately detected by many different methods. The difficulty is that mechanical vibrations and sounds in particular are often very unstable. Only over very short intervals is the signal stable enough to be approximated by a combination of pure frequency tones. We as humans very often recognize sounds as having a particular pitch, a particular color resulting from overtone frequency combinations, and a particular loudness. Yet we also track these qualities as they changexe2x80x94sometimes very rapidlyxe2x80x94with time. No current state of the art technique can even approximate this.
Therefore, a new method of analyzing signals and extracting information from the signals is needed.
Advanced frequency domain methods determine the precise frequency, amplitude, and phase of many components over an interval of time that is short enough to permit the tracking of changing signals. In the case of sound, tracking speed is on the same order as that of a human listener. Tracking is the ability to determine a complete and precise actual, nearly instantaneous spectrum based on a very short time interval. Tying together a sequence of such short intervals produces a profile of the frequency components with time. Moreover, components may be spaced closely together in frequency, subject to minor constraints that rarely compromise the useful practical analysis of real signals.
The embodiments described herein are not limited to mechanical vibrations, but are equally applicable to signals from other sources. That is, the methods may be applied to any system where a spectrum analysis or other transform analysis is employed to seek out signals known to be comprised of, or approximated by, weighted sums of components of known characterization. In addition to sounds and other mechanical vibrations, the signals equally can be any of the varieties of signal, including electrical signals, electromagnetic signals, and spatial signals such as those defined in two dimensions by scanned images. The methods are particularly useful when the need is to extract systems of precise frequency components or other characterizable components, and when such determinations must be accomplished using short sample intervals.
One embodiment pertains to a method of extracting information from a signal, the signal having properties which vary with respect to time. The signal includes a finite number of component signals, the component signals in combination defining the signal. The method analyzes the signal to determine the component signals. The signal analysis includes defining a signal analysis sample interval, the signal analysis sample interval being within a boundary, wherein at least one component signal is included within the signal analysis sample interval. The method transforms the signal over the signal analysis sample interval using a transform method to produce a transform of the signal consisting of transform coefficients. The method fits a function having real and imaginary parameters to a number of transform coefficients to determine a best fit function within the signal analysis sample interval. A mathematical operation performed on the best fit function yields parameter information within the signal analysis sample interval. The method extracts at least one parameter resulting from the mathematical operation, the parameter representing a characteristic of a component signal.
Another embodiment pertains to a method of extracting information from a signal. The method converts the signal to a digital format to produce a digitized signal having a number of digital points. A number of the digital points are sampled to produce a digital sample. The method transforms the digital sample to produce a transform of the signal having a number of transform coefficients. The method samples a number of the transform coefficients to produce a transform sample. The method determines signal information from the transform sample by assuming a number of shapes characterize the transform sample, wherein each shape has a characteristic mathematical relation. The method performs a test of statistical significance on the transform sample using one or more shapes, and outputs signal information based on the test of statistical significance.
Yet another embodiment pertains to a method of determining signal information from a signal. The signal is digitized to produce a digital signal. The method samples the digital signal at intervals to produce signal samples and transforms the signal samples to produce a transformed signal. The method fits one or more functions to the transformed signal, each function having real and imaginary components. Signal information is output based on the result of the fit, the signal information having a relation to a ratio of a real to an imaginary component of a fit function.